Finite Group Databasebeta

← #415797

Subgroups — #415797

Order 512.

2030 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#475C211yesmaximal, characteristic, complemented
2C2×#2547C211yesmaximal, characteristic
3C2×#2538C211yesmaximal, characteristic
4C2×#1990C211yesmaximal, characteristic, complemented
5C2×#1954C211yesmaximal, characteristic
6C2×C4×#103C211yesmaximal, characteristic, complemented
7#11095C214yesmaximal, complemented
8#10970C214yesmaximal, complemented
9#10849C211yesmaximal, characteristic, complemented
10C22×C4⋊Q8C2211yescharacteristic
11C22×#124—21no—
12C22×C22⋊Q8C2212yes—
13C22×C4×Q8C2211yescharacteristic
14C22×C4×D4C2211yescharacteristic
15C2×#475—24no—
16C2×#475—24no—
17C2×#475—24no—
18C22×#106C2212yes—
19C22×C4⋊C8C2211yescharacteristic
20C22×#103C2212yes—
21C22×#103C2212yes—
22C22×C22⋊C8C2214yes—
23C22×C4×C8C2211yescharacteristic
24#2547—24no—
25#2547—24no—
26#2538—24no—
27#2538—24no—
28C2×#397—22no—
29C2×#396—22no—
30C2×#391—21no—
31C2×#390—21no—
32C2×#389—21no—
33C2×#384C2211yescharacteristic
34C2×#380C2212yes—
35C2×#380C2214yes—
36C2×C4×C4⋊C4C2211yescharacteristic
37#1990—24no—
38#1954—24no—
39C4×#103—24no—
40C2×#340C2214yes—
41C2×#340C2212yes—
42C2×#340C2214yes—
43C23×Q8—21no—
44C23×Q8C2311yescharacteristic
45C23×D4D411yescharacteristic
46C24×C4C2312yes—
47C23×C8C2312yes—
48C23×C8C2312yes—
49C2×C4⋊Q8—22no—
50C2×C4⋊Q8—22no—
51C2×C4⋊Q8—22no—
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8D414yes—
55C2×#124—28no—
56C2×#124—44no—
57C2×#124—44no—
58C2×C22⋊Q8—24no—
59C2×C22⋊Q8—24no—
60C2×C22⋊Q8—28no—
61C2×C22⋊Q8—44no—
62C2×C4×Q8—22no—
63C2×C4×Q8—22no—
64C2×C4×Q8—22no—
65C2×C4×Q8—22no—
66C2×C4×Q8—22no—
67C2×C4×Q8D414yes—
68C2×C4×D4—22no—
69C2×C4×D4—22no—
70C2×C4×D4—22no—
71C2×C4×D4—22no—
72C2×C4×D4—22no—
73C2×C4×D4—22no—
74C22×C4⋊C4—21no—
75C22×C4⋊C4—22no—
76C22×C4⋊C4—21no—
77C22×C4⋊C4—21no—
78C22×C4⋊C4—21no—
79C22×C4⋊C4D411yescharacteristic
80C22×C4⋊C4C2312yes—
81C22×C4⋊C4C2312yes—
82C22×C4⋊C4C2311yescharacteristic
83C22×C4⋊C4C2312yes—
84C22×C22⋊C4—22no—
85C22×C22⋊C4—24no—
86C22×C22⋊C4—22no—
87C22×C22⋊C4D412yes—
88C22×C42—21no—
89C22×C42—21no—
90C22×C42C2311yescharacteristic
91#475—232no—
92C2×#106—24no—
93C2×#106—24no—
94C2×#106—24no—
95C2×C4⋊C8—22no—
96C2×C4⋊C8—22no—
97C2×C4⋊C8D414yes—
98C2×#103—24no—
99C2×#103—24no—
100C2×#103—24no—
101C2×#103—24no—
102C2×#103—24no—
103C2×#103—24no—
104C2×C22⋊C8—28no—
105C2×C22⋊C8—28no—
106C2×C22⋊C8—44no—
107C2×C4×C8—22no—
108C2×C4×C8—22no—
109C2×C4×C8—22no—
110#397—216no—
111#396—216no—
112#391—28no—
113#390—28no—
114#389—28no—
115#384—22no—
116#384D414yes—
117#380—24no—
118#380—24no—
119#380—28no—
120#380—28no—
121C4×C4⋊C4—22no—
122C4×C4⋊C4D414yes—
123C2×#95—22no—
124C2×#95—21no—
125C2×#95—24no—
126#340—28no—
127#340—24no—
128#340—44no—
129C25C2×D412yes—
130C22×Q8—24no—
131C22×Q8—21no—
132C22×Q8—22no—
133C22×Q8—21no—
134C22×Q8—21no—
135C22×Q8—21no—
136C22×Q8—21no—
137C22×Q8—22no—
138C22×Q8—22no—
139C22×Q8—42no—
140C22×Q8—42no—
141C22×Q8—42no—
142C22×Q8—42no—
143C22×Q8—42no—
144C22×Q8—42no—
145C22×Q8C2×D412yes—
146C22×Q8#4114yes—
147C22×D4—22no—
148C22×D4—22no—
149C22×D4—22no—
150C22×D4—22no—
151C22×D4—22no—
152C22×D4—22no—
153C22×D4SD1614yes—
154C23×C4—21no—
155C23×C4—22no—
156C23×C4—22no—
157C23×C4—21no—
158C23×C4—22no—
159C23×C4—22no—
160C23×C4—22no—
161C23×C4—22no—
162C23×C4—22no—
163C23×C4—22no—
164C23×C4—22no—
165C23×C4—24no—
166C23×C4—24no—
167C23×C4—24no—
168C23×C4—24no—
169C23×C4C2×D412yes—
170C23×C4C2×D411yescharacteristic
171C23×C4C2×D411yescharacteristic
172C23×C4C2411yescharacteristic
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8—24no—
179C22×C8—22no—
180C22×C8—24no—
181C22×C8—22no—
182C22×C8—22no—
183C22×C8C2×D414yes—
184C22×C8#4114yes—
185C4⋊Q8—28no—
186C4⋊Q8—28no—
187C4⋊Q8—44no—
188C4⋊Q8—44no—
189#124—432no—
190C22⋊Q8—416no—
191C22⋊Q8—48no—
192C22⋊Q8—48no—
193C4×Q8—28no—
194C4×Q8—28no—
195C4×Q8—44no—
196C4×Q8—44no—
197C4×D4—44no—
198C4×D4—44no—
199C4×D4—44no—
200C4×D4—44no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—28no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—48no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4—42no—
233C2×C4⋊C4—42no—
234C2×C4⋊C4—82no—
235C2×C4⋊C4C2×D412yes—
236C2×C4⋊C4C2×D414yes—
237C2×C4⋊C4C2×D414yes—
238C2×C4⋊C4C2×D414yes—
239C2×C22⋊C4—28no—
240C2×C22⋊C4—216no—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—28no—
245C2×C22⋊C4—48no—
246C2×C22⋊C4—48no—
247C2×C22⋊C4—48no—
248C2×C22⋊C4—84no—
249C2×C42—24no—
250C2×C42—24no—
251C2×C42—42no—
252C2×C42—42no—
253C2×C42—42no—
254C2×C42—42no—
255C2×C42C2×D412yes—
256C2×C42C2×D412yes—
257C2×C42C2×D412yes—
258C2×C42C2×D412yes—
259C2×C42C2×D412yes—
260C2×C42#4112yes—
261#106—216no—
262C4⋊C8—28no—
263#103—216no—
264#103—216no—
265C22⋊C8—416no—
266C4×C8—44no—
267#95—28no—
268#95—216no—
269#95—24no—
270C24—22no—
271C24—22no—
272C24—22no—
273C24—24no—
274C24—24no—
275C24—22no—
276C24—22no—
277C24—22no—
278C24—24no—
279C24—24no—
280C24C2×SD1614yes—
281C24C22×D411yescharacteristic
282C2×Q8—22no—
283C2×Q8—24no—
284C2×Q8—22no—
285C2×Q8—22no—
286C2×Q8—24no—
287C2×Q8—24no—
288C2×Q8—22no—
289C2×Q8—22no—
290C2×Q8—24no—
291C2×Q8—24no—
292C2×Q8—41no—
293C2×Q8—41no—
294C2×Q8—42no—
295C2×Q8—42no—
296C2×Q8—42no—
297C2×Q8—42no—
298C2×Q8—42no—
299C2×Q8—48no—
300C2×Q8—48no—
301C2×Q8—48no—
302C2×Q8—84no—
303C2×Q8—84no—
304C2×Q8—84no—
305C2×Q8—84no—
306C2×Q8#12414yescomplemented
307C2×D4—28no—
308C2×D4—28no—
309C2×D4—28no—
310C2×D4—44no—
311C2×D4—44no—
312C2×D4—44no—
313C2×D4—44no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—24no—
318C22×C4—22no—
319C22×C4—24no—
320C22×C4—22no—
321C22×C4—24no—
322C22×C4—22no—
323C22×C4—24no—
324C22×C4—22no—
325C22×C4—24no—
326C22×C4—24no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—42no—
330C22×C4—41no—
331C22×C4—42no—
332C22×C4—41no—
333C22×C4—42no—
334C22×C4—44no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—44no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—44no—
354C22×C4—42no—
355C22×C4—44no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—44no—
361C22×C4—44no—
362C22×C4—44no—
363C22×C4—44no—
364C22×C4—44no—
365C22×C4—44no—
366C22×C4—44no—
367C22×C4—44no—
368C22×C4#12012yes—
369C22×C4#12012yes—
370C22×C4#12012yes—
371C22×C4#12012yes—
372C22×C4#12112yes—
373C22×C4#12112yes—
374C22×C4C2×SD1612yes—
375C22×C4#13512yes—
376C22×C4AGL(1,ℤ/8)12yes—
377C22×C4#13712yes—
378C22×C4C22×D411yescharacteristic
379C22×C4C22×D411yescharacteristic
380C22×C4C22×D411yescharacteristic
381C22×C4C22×D411yescharacteristic
382C22×C4C22×D411yescharacteristic
383C22×C4C2×#4111yescharacteristic
384C22×C4C2×#4112yes—
385C22×C4C2×#4112yes—
386C22×C4H(5,𝔽2)12yes—
387C22×C42−1+412yes—
388C2×C8—24no—
389C2×C8—24no—
390C2×C8—24no—
391C2×C8—28no—
392C2×C8—24no—
393C2×C8—42no—
394C2×C8—42no—
395C2×C8—42no—
396C2×C8—42no—
397C2×C8—42no—
398C2×C8—42no—
399C2×C8—44no—
400C4⋊C4—24no—
401C4⋊C4—22no—
402C4⋊C4—22no—
403C4⋊C4—24no—
404C4⋊C4—24no—
405C4⋊C4—24no—
406C4⋊C4—28no—
407C4⋊C4—416no—
408C4⋊C4—41no—
409C4⋊C4—42no—
410C4⋊C4—42no—
411C4⋊C4—41no—
412C4⋊C4—42no—
413C4⋊C4—42no—
414C4⋊C4—42no—
415C4⋊C4—42no—
416C4⋊C4—44no—
417C4⋊C4—48no—
418C4⋊C4—48no—
419C4⋊C4—48no—
420C4⋊C4—84no—
421C22⋊C4—416no—
422C22⋊C4—432no—
423C22⋊C4—48no—
424C22⋊C4—88no—
425C42—48no—
426C42—48no—
427C42#12014yes—
428C42#12114yes—
429C42#12114yes—
430C42#12714yes—
431C23—28no—
432C23—24no—
433C23—24no—
434C23—28no—
435C23—24no—
436C23—44no—
437C23—44no—
438C23—44no—
439C23—44no—
440C23—44no—
441C23—42no—
442C23—44no—
443C23—44no—
444C23—42no—
445C23—42no—
446C23—42no—
447C23—42no—
448C23—42no—
449C23—42no—
450C23—44no—
451C23—44no—
452C23—42no—
453C23—44no—
454C23C2×#12011yescharacteristic
455C23C2×#12011yescharacteristic
456C23C2×#12111yescharacteristic
457C23#54612yes—
458C23#54712yes—
459C23C22×SD1611yescharacteristic
460C23C2×#13511yescharacteristic
461C23C2×AGL(1,ℤ/8)11yescharacteristic
462C23C2×#13711yescharacteristic
463C23#57712yes—
464C23#57812yes—
465Q8—24no—
466Q8—24no—
467Q8—42no—
468Q8—44no—
469Q8—42no—
470Q8—44no—
471Q8—816no—
472D4—416no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—24no—
476C2×C4—24no—
477C2×C4—24no—
478C2×C4—42no—
479C2×C4—44no—
480C2×C4—44no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—44no—
485C2×C4—44no—
486C2×C4—44no—
487C2×C4—44no—
488C2×C4—42no—
489C2×C4—44no—
490C2×C4—42no—
491C2×C4—44no—
492C2×C4—44no—
493C2×C4—44no—
494C2×C4—44no—
495C2×C4—42no—
496C2×C4—44no—
497C2×C4—42no—
498C2×C4—44no—
499C2×C4—84no—
500C2×C4—82no—
501C2×C4—82no—
502C2×C4—82no—
503C2×C4—82no—
504C2×C4—82no—
505C2×C4—82no—
506C2×C4—82no—
507C2×C4—82no—
508C2×C4—84no—
509C2×C4—82no—
510C2×C4—82no—
511C2×C4—82no—
512C2×C4—82no—
513C2×C4—84no—
514C2×C4—82no—
515C2×C4—82no—
516C2×C4—84no—
517C2×C4#39214yes—
518C2×C4#39214yes—
519C2×C4#44814yes—
520C2×C4#44914yes—
521C2×C4#45014yes—
522C2×C4#45214yes—
523C2×C4#46514yes—
524C2×C4#46814yes—
525C2×C4C2×#12011yescharacteristic
526C2×C4C2×#12111yescharacteristic
527C2×C4C2×#12111yescharacteristic
528C2×C4#52512yes—
529C2×C4C2×#12411yescharacteristic
530C2×C4#52912yes—
531C2×C4C2×#12711yescharacteristic
532C2×C4#53211yescharacteristic
533C2×C4#53511yescharacteristic
534C2×C4#53611yescharacteristic
535C2×C4#53712yes—
536C2×C4#54212yes—
537C2×C4#54612yes—
538C2×C4#54612yes—
539C2×C4#54812yes—
540C2×C4#54812yes—
541C2×C4#54812yes—
542C2×C4#55012yes—
543C8—44no—
544C8—44no—
545C22—44no—
546C22—48no—
547C22—44no—
548C22—44no—
549C22—48no—
550C22—84no—
551C22—82no—
552C22—84no—
553C22—82no—
554C22—82no—
555C22—82no—
556C22C2×#39211yescharacteristic
557C22#240712yes—
558C22#242712yes—
559C22#243211yescharacteristic
560C22#243511yescharacteristic
561C22C2×#45011yescharacteristic
562C22C2×#44811yescharacteristic
563C22C2×#44911yescharacteristic
564C22C2×#45211yescharacteristic
565C22#300911yescharacteristic
566C22#301111yescharacteristic
567C22#301312yes—
568C22C2×#46511yescharacteristic
569C22C2×#46811yescharacteristic
570C22#306312yes—
571C22#329812yes—
572C22#330412yes—
573C22#332812yes—
574C22#333212yes—
575C22#333512yes—
576C22#334212yes—
577C22#335312yes—
578C22#335612yes—
579C4—24no—
580C4—44no—
581C4—84no—
582C4—84no—
583C4—84no—
584C4—82no—
585C4—82no—
586C4#200018yes—
587C4#242612yes—
588C4#243111yescharacteristic
589C4#243611yescharacteristic
590C4#254012yes—
591C4#262011yescharacteristic
592C4#285011yescharacteristic
593C2—84no—
594C2#1521511yescharacteristic
595C2C2×#200011yescharacteristic
596C2#2339411yescharacteristic
597C2#2412012yes—
598C2#2412312yes—
599C2#2534112yes—
600C2#2534812yes—
601C2#2570211yescharacteristic
602C2#2570511yescharacteristic
603C2#2588411yescharacteristic
604C2#2588711yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×D4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C22×#475C2semidirect
4. C2×#1990C2semidirect
6. C2×C4×#103C2semidirect
7. #11095C2semidirect
8. #10970C2semidirect
9. #10849C2semidirect
306. C2×Q8#124semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C22×#4752
C2×#25472
C2×#25382
C2×#19902
C2×#19542
C2×C4×#1032
#110952
#109702
#108492